Basic DefinitionsScalars a real number latercomplexNotation CEREg c Z OT 2e Os

Vectors a list of real numbersNota v c IR n length of the vector

size dimension

Eg Eye c IRNde Usually work as a column

can also write 17 ay 2e

Two vectors are equal if they have thesame size and all components are equal

Zero vector O E

µY I write vectors with normal letters UN wOften see I

ons U V w

Scalar Multiplication scalar x vector

CER VEIR cre IR

4 tt skit SVector t yw c IR ut w v WER

itthe same

4 Ea Ea size

Geometry

Draw a vector as an amnioComponents are displacements in different directions

r tss.liScalar multiplication

changes the length can reverse direction

ir CD.li i2v l3d

re

Vector addition Parallelogram lao

a

Vector Subtraction art w v w

quu vector from head of v

e to head of w

Linear Combinations combine additiondescalarmult

Yw ER DEIR w cvtdwc.IR

c dm CEE'Ee Dnew 2utu

i f 2vtwn I i

cutdwgo cx lengthof v

v w in v direction

thendelengthofwin w direction

Dot Product EEEYin Hgs Xcy t taryn c IRvectorvector scalar

Ig I 2 2 1 3 C 37 5

Rules for vector algebrau cw e law cu w

u v V Uor u vtu wvtu w

u lv.ir v.XVw is a scale

Special Case

YL E x x x XE Xi't xi 20

In general v v sum of squares of entries 70

Length of v is E Hrh

HEY KeithSanity Check CHR vetR

kill EL EILex ca t Exa exa

alxet xi o Hull

Ko ult Id killus 2v is twice as long as V

S B Iv

Distance Wuv

a

distance from r to w Hw vkHr wk

Unit Vector a vector v with Hull 1

i.e vector 1 st xie 4 5 1

If u 0 the unit rector in director of v

is u H

NB Hull HEH Ark 1

Eg v 7 Hulk TEE 5

u s TIESAngles what is v w ifwtv Geometrically

c Aw ukLaw of Cesmes ano

c za b Lab cos a a b g

a lb s Ault b 464

Vector versionKu ru Ault'tHalf HullHollcosa ETFu v Lw v w w Zu wt v v

Hw t

tAEYµµ a2kHHullcost

v w Hull Hull cos 0

Cos aVI

HullHull 40 10

In dim this defines

angle from u to w cos IffyIV of Hull Hull loos OIEHH.hu l

Schwartz InequalityIv wt a Hull Hull

Defn Two vectors are orthogonal or perpendicularif v

Says cos E 0 0 190gyu

Tsumum

A matrix is a box holding a grid of numbers

A Es II I 3 ins3,2 entry 3rd row 2ndcol

mxn 3 2 size of the matrix

Addition Scalar multiplication componentwise

ok Et k l's E

KEITH IE'T3same size

NB A vector is a matrix with 1 columnnow vector C 2 3

Matrixx nectar 2 wayscolumn first

m I E fix3 x I tuff c

u.nl i eYaiah n Ix

Row first

E E3 I 53 1 6 2

entries of product are dot productsw the crows

NB Only makes sense if cels of matrixsize of the vector

man att vs m x

NB Recover v w by thinking of u as a

new vector

ti x z as lxiyctxigstxsg.se

MatrixxMatrix also 2 waysBy columns

Alin Yiu LAI HAIBy news

I 7141471 c INB mxn x nxp m xp

teamed

i i

i

n

If I Cuz

rows

i 231 2 723

a aIf EE43

Eg Identity matrix square Gen

Is f In o

O

Checks In A A A In

properties ASSOMIE Dimensions MATCHAssociativity AB C A CBC

wrote ABCGu vector IAB v A Br

DistributivityA Btc ABtAC CAN C ACtBc

Commutativity failsAB BA A 3 2 B 2 4

AB V BAXeven for 2 2 matches

Cancellation failsA B AC A to Be